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Coherence is one of the most important concepts in optics and is strongly related to the ability of light to exhibit interference effects.
A light field is called coherent when there is a fixed phase relationship between the electric field values at different locations or at different times.
Partial coherence means that there is some (although not perfect) correlation between phase values.
There are various ways of quantifying the degree of coherence, as described below.

It is also common to call certain processes or techniques coherent or incoherent.
In that case, “coherent” essentially means phase-sensitive.
For example, the general method of coherent beam combining relies on the mutual coherence of beams, whereas spectral (incoherent) beam combining does not

Spatial Versus Temporal Coherence

Figure 1:
A prism is inserted into a spatially coherent laser beam, generating an interference pattern on the screen.

There are two very different aspects of coherence:

Spatial coherence means a strong correlation (fixed phase relationship) between the electric fields at different locations across the beam profile.
For example, within a cross-section of a beam from a laser with diffraction-limited beam quality, the electric fields at different positions oscillate in a totally correlated way, even if the temporal structure is complicated by a superposition of different frequency components.
Spatial coherence is the essential prerequisite of the strong directionality of laser beams.

Temporal coherence means a strong correlation between the electric fields at one location but different times.
For example, the output of a single-frequency laser can exhibit a very high temporal coherence, as the electric field temporally evolves in a highly predictable fashion: it exhibits a clean sinusoidal oscillation over extended periods of time.

Lasers have the potential for generating beams (e.g. Gaussian beams) with very high spatial coherence, and this is perhaps the most fundamental difference between laser light and radiation from other light sources.
High spatial coherence arises from the existence of resonator modes, which define spatially correlated field patterns.
In situations where only a single resonator mode has sufficient laser gain to oscillate, a single longitudinal mode can be selected, obtaining single-frequency operation with very high temporal coherence as well.

Figure 2:
Electric field distribution around the focus of a Gaussian laser beam (coming e.g. from the left side) with perfect spatial and temporal coherence.

Figure 3 shows a beam with high spatial coherence, but poor temporal coherence.
The wavefronts are formed as above, and the beam quality is still very high, but the amplitude and phase of the beam varies along the propagation direction.
Note that both the local amplitude and the spacing of the wavefronts vary to some extent.
Such a beam can be generated e.g. from the output of a supercontinuum source.

Figure 3:
A laser beam with high spatial coherence, but poor temporal coherence.

Figure 4 shows a laser beam with reduced spatial coherence, but high temporal coherence.
The wavefronts are deformed, and this results in a high beam divergence and poor beam quality.
On the other hand, the beam is monochromatic, so that the spacing of the deformed wavefronts remains constant.
Such a beam can result from a single-frequency laser, when its output is sent through some optically inhomogeneous material.

Figure 4:
A laser beam with poor spatial coherence, but high temporal coherence.

If a laser beam with high spatial coherence is sent through an optical element (for example, a very inhomogeneous piece of glass) which totally scrambles the wavefront, the resulting distorted beam could in principle still be considered as spatially coherent, since the phase relationship between the electric fields at different points would still be fixed, as long as the wavefront distortions do not change with time.
It would then also in principle be possible to restore a simple beam shape by applying another optical element which compensates the complicated spatial distortions.
For most practical purposes, however, such a distorted beam will be spatially incoherent.
Nevertheless, a real destruction of special coherence requires time-dependent wavefront distortions, as can be obtained with a rotating diffuser, for example.
In that case, the temporal coherence is also somewhat degraded.

Quantifying Coherence

There are different ways to quantify the degree of coherence:

Correlation functions specify the degree of correlation as a function of a spatial or temporal distance.
There are correlation functions of different orders.
The first-order correlation function is related to the optical spectrum.
The second-order correlation function describes intensity correlations, i.e., effects like photon bunching or anti-bunching.
Higher-order functions describe still more subtle details.

The coherence time quantifies the degree of first-order temporal coherence via the time over which coherence is lost.

For other applications, the coherence of the light used should be as low as possible.
For example, very low temporal coherence (but combined with high spatial coherence) is required for coherence tomography, where images are created with a kind of interferometry, and a high spatial resolution requires low temporal coherence.
Suitable light sources for such applications can be based on amplified spontaneous emission (ASE) from a laser amplifier (→ superluminescent sources) or on supercontinuum generation in nonlinear media.
A low degree of temporal coherence can also be beneficial for laser projection displays, imaging and pointer applications, as it reduces the tendency for speckle and similar interference effects.

Coherence in Quantum Optics

In quantum optics, the term coherence is often used for the state of light-emitting atoms or ions.
In that case, coherence refers to a phase relationship between the complex amplitudes corresponding to electronic states.
This is important, e.g., in the context of lasing without inversion.
There is also the term “coherent states” of the light field, which has yet another meaning.